A309148 A(n,k) is (1/k) times the number of n-member subsets of [k*n] whose elements sum to a multiple of n; square array A(n,k), n>=1, k>=1, read by antidiagonals.
1, 1, 0, 1, 1, 1, 1, 2, 4, 0, 1, 3, 10, 9, 1, 1, 4, 19, 42, 26, 0, 1, 5, 31, 115, 201, 76, 1, 1, 6, 46, 244, 776, 1028, 246, 0, 1, 7, 64, 445, 2126, 5601, 5538, 809, 1, 1, 8, 85, 734, 4751, 19780, 42288, 30666, 2704, 0, 1, 9, 109, 1127, 9276, 54086, 192130, 328755, 173593, 9226, 1
Offset: 1
Examples
Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, 6, ... 1, 4, 10, 19, 31, 46, 64, ... 0, 9, 42, 115, 244, 445, 734, ... 1, 26, 201, 776, 2126, 4751, 9276, ... 0, 76, 1028, 5601, 19780, 54086, 124872, ... 1, 246, 5538, 42288, 192130, 642342, 1753074, ...
Links
- Alois P. Heinz, Rows n = 1..150, flattened
Crossrefs
Programs
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Maple
with(numtheory): A:= (n, k)-> add(binomial(k*d, d)*(-1)^(n+d)* phi(n/d), d in divisors(n))/(n*k): seq(seq(A(n, 1+d-n), n=1..d), d=1..12); -
Mathematica
A[n_, k_] := 1/(n k) Sum[Binomial[k d, d] (-1)^(n+d) EulerPhi[n/d], {d, Divisors[n]}]; Table[A[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 04 2019 *)
Formula
A(n,k) = 1/(n*k) * Sum_{d|n} binomial(k*d,d)*(-1)^(n+d)*phi(n/d).
A(n,k) = (1/k) * A304482(n,k).
Comments